Stationary phase analysis for analytic newvectors and application to subconvexity problems
Liyuan Ye
TL;DR
The paper advances subconvexity bounds for triple product and Rankin–Selberg L-functions in a hybrid spectral and level framework by resolving the archimedean component of a local test-vector conjecture. It develops a robust stationary phase analysis for analytic newvectors on PGL2(R) and PGL2(C), connecting their oscillatory Whittaker behavior to explicit conductor control. By combining these archimedean inputs with a refined regularized triple product formula and an amplification strategy, the authors prove a bound L(π1⊗π2⊗π3,1/2) ≪ C(π1⊗π2)^{1/2+ε} (C(π1⊗π2)/C(π2⊗π2))^{−δ}, with δ>0 when π1⊗π2 stays away from QUE-like cases; in particular, a subconvex bound arises in the QUE-avoiding regime. The results generalize HMN22 to archimedean places and across number fields, offering a powerful framework for subconvexity that handles conductor dropping and yields new hybrid bounds via period methods.
Abstract
In this paper, we extend the results of Michel-Venkatesh and Hu-Michel-Nelson to establish an upper bound for triple product and Rankin-Selberg L-functions of the form $$L(π_1 \otimes π_2 \otimes π_3,\frac{1}{2})\ll_{π_3,ε}C(π_1\otimesπ_2)^{\frac{1}{2} + ε} \left( \frac{C(π_1 \otimes π_2)}{C(π_2 \otimes π_2)}\right)^{-δ}$$ in the spectral aspect, allowing conductor dropping. In particular, we obtain a subconvexity bound when $π_1\otimesπ_2$ stays uniformly away from QUE-like case. The new ingredient is a stationary phase analysis of the analytic newvectors introduced by Jana and Nelson in \cite{JN19}, for both $\mathrm{PGL}_2(\mathbb{R})$ and $\mathrm{PGL}_2(\mathbb{C})$, which is applied to a test vector conjecture for local triple product periods.